In this research article, we present several generalizations of Qi’s inequality on time scales. We establish dynamic versions of Callebaut’s inequality and Cauchy-Schwarz’s inequality on time scales. To establish our results, we apply the diamond-alpha integral and the time scale $∆$ or $∇$-Riemann-Liouville type fractional integrals. Our findings unify and extend discrete, continuous and quantum analogues.

In this study, we introduce a novel hybrid identity that successfully combines Newton-Cotes and Gauss quadratures, enabling us to recover both Simpson’s second formula and the left and right Radau 2 point rules, among others. Based on this versatile foundation, we establish some new biparametric fractional integral inequalities for functions whose first derivatives are extended $s$-convex in the second sense. To support our findings, we present illustrative examples featuring graphical representations and conclude with several practical applications to demonstrate the effectiveness of our results.

This paper is mainly concerned with the existence and simulation of solutions for a class of Caputo fractional differential systems with $p$-Laplacian operators on star graphs. The Hyers-Ulam stability of the systems on star graphs is also proved. Furthermore, an example on a formaldehyde graph is presented to demonstrate the practicality of the main results. The innovation of this paper lies in combining a fractional differential system with a formaldehyde graph, interpreting the chemical bonds as the edges of the graph, and exploring the existence and numerical simulation of solutions to the fractional differential system on this unique graph structure.

This paper introduces a unique strategy for solving numerically a class of nonlinear Pantograph differential equations using the Fibonacci wavelet collocation method (FWCM). First, we transform the nonlinear Pantograph differential equations system into a nonlinear algebraic system using this proposed approach. Next, the transformed nonlinear algebraic system is solved by using the Newton-Raphson scheme. The main advantage of this approach lies in its ability to reduce the computational complexity associated with solving Pantograph equations, resulting in accurate and efficient solutions. Comparative analyses with other established numerical methods reveal its superior accuracy and convergence rate performance. Further, a few examples are provided to evaluate the effectiveness of the suggested approach using absolute error functions. As far as our literature survey indicates, no one attempted the nonlinear Pantograph differential equations by FWCM. It compels us to study a system of Pantograph differential equations via FWCM.

In this paper, we investigate the entire solutions for a certain class of non-linear difference equations of the form: $f^n+ q(z)e^{Q(z)}\mathcal{L}_1(z, f)$ $= α_1(z)e^{β_1(z)}+α_2(z)e^{ β_2(z)},$ where $\mathcal{L}_1(z, f)$ is the generalized linear difference operator, $α_1(z)$ and $α_2(z)$ are non-zero small functions of $f,$ $q(z)$ and $Q(z)$(nonconstant), $β_1(z)$ and $β_2(z)$ are non-zero polynomials. Our results improve upon and generalize some previously established findings.

In this manuscript, we study the split null point problem in the settings of real Hilbert spaces using two different iterative methods. In our first method, we propose a self-adaptive algorithm with an inertial technique for solving split common null point problem and fixed point of a finite family of a demimetric mapping without the computation of the resolvent of a monotone operator. In our second method, we propose a self-adaptive algorithm with a multi-step inertial technique to approximate a solution of the aforementioned problems and to accelerate the rate of convergence of our iterative method. The selection of the stepsize employed in our iterative algorithms does not require prior knowledge of the operator norm. Lastly, we present a numerical example to show the performance of our iterative algorithms. The result discussed in this article extends and complements many related results in literature.

The ongoing COVID-19 pandemic, caused by the highly contagious coronavirus, poses significant challenges to public health worldwide. Effective control measures are essential to mitigate the spread of the virus and protect vulnerable populations. This study aims to develop novel mathematical models using fractional derivatives to analyze the dynamics of the COVID-19 outbreak. By employing modified mathematical procedures, we explore the impact of quarantine and isolation as control measures on the disease’s transmission dynamics. We investigate a system representing COVID-19 through three different arbitrary-order derivative operators: the Atangana-Baleanu derivative with the generalized Mittag-Leffler function, the Caputo derivative with a power law, and the Caputo-Fabrizio derivative with exponential decay. Using fixed-point theory, we assess the existence and uniqueness of solutions for the arbitrary-order system. Our analysis includes numerical simulations that reveal how varying the fractional order influences the behavior of the epidemic. The results demonstrate that increasing the fractional order generally slows the disease’s progression, reflecting the memory effect inherent in fractional derivatives. Specifically, higher values of the fractional order correspond to a more gradual spread, reducing the peak number of infections and extending the outbreak’s duration. The work highlights the critical importance of using fractional order models to capture the complex dynamics of disease spread and emphasizes that the implementation of quarantine and isolation for treatment significantly decreases the cumulative number of new cases and the overall transmission rate of COVID-19. This research underscores the effectiveness of utilizing fractional-order models to better understand and control the complex dynamics of disease transmission.

This work presents an approach for solving the split feasibility problem in an efficient manner. For solving the split feasibility problem, we present a method with a two-step inertial extrapolation and self-adaptive stepsize. The adjustable stepsize and two-step inertial extrapolation both contribute to the proposed method’s improved rate of convergence and decreased computational complexity. The strong convergence results are obtained without on-line rule of the inertial parameters and the iterates. This makes our proof arguments different from what is obtainable in the literature where online rule is imposed on algorithms involving inertial extrapolation step. As far as we know, no strong convergence result has been obtained before now for algorithms with two step inertial for solving split feasibility problems in the literature. To demonstrate the viability of our suggested strategy, numerical results are provided at the end.

In this paper, we study the following Navier problem

Here, $h ∈ L^{p'} (\mathcal{Q}, v^{1−p′}_1),$ $\mathcal{K}, \mathcal{L}$ and $b$ are Carathéodory functions and $ϕ_1,ϕ_2,$ $v_1, v_2, v_3$ and $v_4$ are $A_p$-weights functions. By using the theory of monotone operators (Browder–Minty Theorem), we demonstrate the existence and uniqueness of weak solution to the above problem.

The differential transformation method (DTM) and Adomian decomposition method (ADM) are two numerical methods that can be used to solve various differential equations. In this work, we compare the accuracy, convergence, and computational complexity of these two methods by using them to solve second-order nonlinear equations using a new differential operator for the second-order equation. We also used both methods to solve second-order nonlinear differential equations.

In this paper, we establish the existence and uniqueness of mild solutions for the time fractional hall-magneto-hydrodynamics stochastic equations with a fractional derivative of Caputo. Initially, we focus on the existence and uniqueness in the deterministe case. Using the Mittag-Leffler operators $\{\mathcal{Q}_α(−t^α \mathbb{J}): t ≥ 0\}$ and $\{\mathcal{Q}_{α,α}(−t^α \mathbb{J}) : t ≥ 0\}$ and applying the bilinear fixed-point theorem, we will prove the frist result. Next, by Itô integral, and by similair analogy we will establish the existence and uniqueness in the stochastic case in $\mathcal{EN}^{\mu}_a ∩ N^{2α}_{a,\mu}.$

In this paper, we introduce a new inertial type algorithm with a self-adaptive step size for approximating a common element of the set of solutions of split common null point and pseudomonotone variational inequality problem as well as the set of common fixed point of a finite family of quasi nonexpansive mappings in uniformly smooth and 2-uniformly convex real Banach space. The proposed algorithm is constructed in such a way that its convergence analysis does not require a prior estimate of the operator norm. We also give numerical examples to illustrate the performance of our algorithm. Our results generalize and improve many existing results in the literature.

In this paper, we are interested in some results of the existence of multiple solutions for Navier boundary value problem involving degenerated $(p(.), q(.))$-Biharmonic and $(p(.), q(.))$-Laplacian operators. Our approach is based on variational method and critical point theory.

In this paper, we propose a class of general non-polynomial analytic oscillator models, and study the limit cycle bifurcation at the nilpotent singularity or elementary center-focus. By Taylor expansion, two specific systems from the original model are transformed into two equivalent infinite polynomial systems, and the highest order of fine focus as the nilpotent Hopf bifurcation or Hopf bifurcation point is determined respectively. At the same time, the local bifurcation cyclicities and center problems for two systems are solved respectively. To our knowledge, such dynamic properties are rarely analyzed in many non-polynomial models.

The Steiner distance for the set $S ⊆ V (G)$ would simply be the number of edges in the minimal subtree connecting them and is denoted as $d_G(S).$ The Steiner-Harary index is $SH_k(G),$ defined as the sum of the reciprocal of the Steiner distance for all subsets with $k$ vertices in $G.$ In this article, we calculate the exact value of $SH_k(G)$ for specific graphs and establish new best possible lower and upper bounds and characterization. Furthermore, we explore the relationship between $SH_k(G)$ and other graph indices based on Steiner distance.

We study the initial-boundary value parabolic problem involving variable exponent under the generalized Leray-Lions conditions. We clarify the definitions of entropy and renormalized solutions to such parabolic problems, and we establish the equivalence between these definitions of entropy and renormalized solutions to the parabolic problems with the Leray-Lions operator and with measure data.

This paper investigates a class of fractional problems involving the variable-order $p(·, ·)$-Laplacian with homogeneous Dirichlet boundary conditions. Under suitable assumptions on the nonlinear term, we establish novel existence and uniqueness results for weak solutions. We achieve this by combining variational techniques with a result from the theory of monotone operators. Additionally, we reveal several interesting properties of the solution.

This paper investigates a frictional contact problem involving a thermo-electro-visco-elastic model for locking materials in contact with a rigid foundation. Friction is described by the subgradient of a locally Lipschitz function, while contact is governed by Signorini’s unilateral condition. We formulate the problem as a system of three hemivariational inequalities and establish an existence and uniqueness theorem using a fixed-point argument and recent advances in hemivariational inequalities theory. Finally, we present a fully discrete finite element approximation of the model and derive error estimates for the approximate solution.

In the paper, we study the question about global unsolvability of the Kawahara and Kaup-Kupershmidt shallow water equations in a bounded domain. For certain initial-boundary-value problems of the shallow-water equations, we establish the necessary conditions for the existence of global solutions. The proof of the results is based on the nonlinear capacity method. In closing, we provide some examples.

This is the last part of our work series on a class of cooperative reaction-diffusion systems with free boundaries in one space dimension, where the diffusion terms are nonlocal, given by integral operators involving suitable kernel functions, and some of the equations in the system do not have a diffusion term. Such a system covers various models arising from population biology and epidemiology, including in particular a West Nile virus model [10] and some epidemic models [22, 38], where a “spreading-vanishing” dichotomy is known to govern the long time dynamical behaviour, but the spreading rate was not well understood. In this work series, we develop a systematic approach to determine the spreading profile of the system. In Part 1 [11], we obtained threshold conditions on the kernel functions which decide exactly when the spreading has finite speed $c_0,$ or infinite speed (accelerated spreading), and for the case of finite speed, we determined its value $c_0$ via semi-wave solutions. In this paper, for some typical classes of kernel functions, we obtain more precise descriptions of the spreading for the finite speed case by revealing the exact rate of shifting of the spreading front from $c_0t;$ the infinite speed case is studied separately in Part 2 [14].